Let ư, v, w e R* be three nonzero vectors and { u, v} be a linearly independent set. Then which of the following is always correct ? I. If w 4 (, v), then the set {u, v, w} is linearly independent. II. If ve (7, w), then the set { u + v,ữ + w, v - w} is linearly independent. III. If T £ (7, w), then the set {u, v, w} spans R. IV. For all 3 e R?, E (T,V). (a) Only II (b) Only I (c) Only II and II (d) Only II and Iv (e) Only I, III and IV

Let ư, v, w e R* be three nonzero vectors and { u, v} be a linearly independent set. Then which of the following is always correct ? I. If w 4 (, v), then the set {u, v, w} is linearly independent. II. If ve (7, w), then the set { u + v,ữ + w, v - w} is linearly independent. III. If T £ (7, w), then the set {u, v, w} spans R. IV. For all 3 e R?, E (T,V). (a) Only II (b) Only I (c) Only II and II (d) Only II and Iv (e) Only I, III and IV

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let ư, v, w e R* be three nonzero vectors and { u, v} be a linearly independent set. Then which
of the following is always correct ?
I. If w 4 (T, v), then the set {u, v, w} is linearly independent.
II. If ve (7, w), then the set { u + v,ữ + w, v - w} is linearly independent.
III. If T (V, w), then the set { u, v, w} spans R.
IV. For all 3 e R?, E (T,V).
(a) Only II
(b) Only I
(c) Only II and II
(d) Only II and IV
(e) Only I, III and IV

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